Predictive Model for Use in Sequencing-by-Synthesis

ABSTRACT

A method of obtaining a more accurate estimate of a signal correction parameter(s) in sequencing-by-synthesis operations, such as incomplete extension rates, carry forward rates, and/or signal droop rates. The sequencing operation produces signal data. A model is constructed to simulate a population of template strands as it undergoes the sequencing process and becomes divided into different phase-states as the sequencing-by-synthesis progresses. For example, the model may be a phase-state model. The output from the model is used to adjust the signal correction parameter(s). For example, the model may be fitted to the signal data. This fitting results in a more accurate estimate of the signal correction parameter(s). In another embodiment, the signal droop rate is modeled as a decaying function and this decaying function is fitted to the signal data to obtain an improved estimate of the signal droop rate.

RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser. No. 61/407,377 filed on 27 Oct. 2010, which is incorporated by reference herein in its entirety.

TECHNICAL FIELD

The present application generally relates to nucleic acid sequencing, and more particularly, to a statistical model to estimate signal correction parameters.

BACKGROUND

Sequencing-by-synthesis is among a new generation of high throughput DNA sequencing technologies. Examples of techniques and platforms for sequencing-by-synthesis include the Genome Analyzer/HiSeq/MiSeq platforms (Illumina, Inc.; see, e.g., U.S. Pat. Nos. 6,833,246 and 5,750,341); those applying pyrosequencing-based sequencing methods such as that used by Roche/454 Technologies on the GS FLX, GS FLX Titanium, and GS Junior platforms (see, e.g., Ronaghi et al., SCIENCE, 281:363 (1998) and Margulies et al., NATURE, 437:376-380 (2005)); and those by Life Technologies Corp./Ion Torrent in the PGM™ system (see, e.g., U.S. Patent Application Publication Nos. 2010/0137143 and 2009/0026082, which are incorporated by reference in their entirety). As will be further explained below, one of the problems in sequencing-by-synthesis is the loss of phase synchrony and/or signal droop, which can hinder the ability to make accurate base calls. More accurate estimates of the phasing effects and/or signal droop can improve the ability to make accurate base calls.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings illustrate one or more exemplary embodiments of the present invention and serve to explain the principles of various exemplary embodiments. The drawings are exemplary and explanatory only and are not in any way limiting of the present invention.

FIG. 1 shows a DNA sample undergoing a sequencing reaction.

FIG. 2 shows an example of an ionogram from which base calls can be made.

FIG. 3A shows an example of an ionogram for a sequencing read prior to signal correction for phasing effects. FIG. 3B shows an ionogram for the same sequencing read after signal correction for phasing effects.

FIG. 4 shows a flow chart illustration of an embodiment of the present invention.

FIG. 5 shows a flow chart illustration of another embodiment of the present invention.

FIG. 6 shows an example of a sequencing-by-synthesis operation.

FIG. 7 is a diagram showing a sequencing apparatus according to an embodiment of the present invention.

FIG. 8 shows an apparatus according to an embodiment of the present invention.

EXEMPLARY EMBODIMENTS

In an embodiment, the present invention provides a method of sequencing a polynucleotide strand, comprising: (a) flowing a series of nucleotides to the polynucleotide strand; (b) obtaining signal data relating to chemical reactions resulting from the flow of nucleotides; (c) determining, using the signal data, sequence information of at least a portion of the polynucleotide strand; (d) constructing a model for a set of flows that encompasses the sequence information, wherein said model includes a signal correction parameter; (e) calculating, using the phase-state model, predicted signals resulting from the set of nucleotide flows; (f) comparing the predicted signals to the signal data; and (g) adjusting the signal correction parameter of the phase-state model based on the comparison of the predicted signals to the signal data. In some cases, the model is a phase-state model that simulates a population of the polynucleotide strands.

In some cases, the nucleotides are flowed onto an array having multiple wells, wherein the polynucleotide strand is contained in a first well of the array, and the method further comprises: obtaining signal data relating to chemical reactions in a plurality of other wells within a region around the first well; and performing steps (c) through (g) for each of the obtained signal data from the plurality of other wells to obtain multiple adjusted signal correction parameters. In some cases, the method further comprises calculating a region-wide estimate of the signal correction parameter using the multiple adjusted signal correction parameters. In some cases, the comparing step comprises calculating a fitting metric that measures the fit between the predicted signals and the signal data from at least some of the plurality of wells. In some cases, the fitting metric measures the fit between the predicted signals and the signal data from less than all of the plurality of wells; wherein the region-wide estimate excludes adjusted signal correction parameters from wells that produce a fitting metric exceeding a predetermined threshold. In some cases, the method further comprises performing a base calling analysis of the signal data from multiple wells within the region using the region-wide estimate of the signal correction parameter.

In some cases, the method further comprises repeating steps (d) through (g) using the adjusted signal correction parameter. In some cases, use of the adjusted signal correction parameter improves the fit between the signal data and the predicted signals. In some cases, calculating the predicted signals uses a signal droop rate as one of the terms. In some cases, the phase-state model includes two or more signal correction parameters, including a carry forward rate and an incomplete extension rate.

In some cases, the comparing step comprises calculating a fitting metric that measures a fit between the predicted signals and the signal data. In some cases, the adjusting step comprises determining a value of the signal correction parameter that optimizes the fitting metric. In some cases, the fitting metric is calculated using only nucleotide flows that result in nucleotide non-incorporation or single nucleotide incorporations. In some cases, the method further comprises performing a base calling analysis of the signal data using the adjusted signal correction parameter.

In some cases, the set of nucleotide flows is a first set of nucleotide flows and the sequence information is a first sequence information, and the method further comprises: applying the phase-state model using the adjusted signal correction parameter; calculating, using the phase-state model, predicted signals resulting from a second set of nucleotide flows that includes nucleotide flows that are not in the first set of nucleotide flows; making base calls by comparing the signal data to the predicted signals; and obtaining a second sequence information about the polynucleotide strand, wherein the second sequence information includes sequence information not contained in the first sequence information. In some cases, the method further comprises repeating steps (d) through (g) using the second sequence information to obtain a further adjusted signal correction parameter.

In some cases, the array includes a chemFET sensor array for detecting a reaction of the nucleotides with the contents of the wells in the array. In some cases, the phase-state model simulates a population of the polynucleotide strands. In some cases, the calculation of the predicted signals includes the use of a signal droop rate. In some cases, the phase-state model is adjusted for the signal droop rate.

In some cases, the region is a first region and the signal droop rate is obtained by a method comprising: receiving signal data relating to chemical reactions in a plurality of wells within a second region of the array, wherein the plurality of wells includes the well containing the polynucleotide strand, wherein the second region is the same or different from the first region; calculating a set of averaged signal values from the signal data; and determining a region-wide signal droop rate by fitting a signal decay function to the set of averaged signal values.

In another embodiment, the present invention provides a non-transitory machine-readable storage medium comprising instructions which, when executed by a processor, cause the processor to perform the above-described methods. In some cases, the processor is caused to: (a) receive signal data relating to chemical reactions resulting from a flow of nucleotides onto an array containing multiple wells, at least one of said wells containing a polynucleotide strand; (b) determine, using the signal data, sequence information of at least a portion of the polynucleotide strand; (c) construct a phase-state model stored in a computer memory, wherein said model is constructed for a set of flows that encompasses the sequence information and includes a signal correction parameter; (d) calculate, using the phase-state model, predicted signals resulting from the set of nucleotide flows; (e) compare the predicted signals to the signal data; (f) adjust the signal correction parameter of the phase-state model based on the comparison of the predicted signals to the signal data; and (g) store the adjusted signal correction parameter in the memory.

In another embodiment, the present invention provides an apparatus comprising: a machine-readable memory; and a processor configured to execute machine-readable instructions, said instructions which when executed cause the apparatus to perform the above-described methods. In some cases, the apparatus is programmed to: (a) receive signal data relating to chemical reactions resulting from a flow of nucleotides onto an array containing multiple wells, at least one of said wells containing a polynucleotide strand; (b) determine, using the signal data, sequence information of at least a portion of the polynucleotide strand; (c) construct a phase-state model stored in the memory, wherein said model is constructed for a set of flows that encompasses the sequence information and includes a signal correction parameter; (d) calculate, using the phase-state model, predicted signals resulting from the set of flows; (e) compare the predicted signals to the signal data; (f) adjust the signal correction parameter of the phase-state model based on the comparison of the predicted signals to the signal data; and (g) store the adjusted signal correction parameter in the memory. In some cases, the apparatus further comprises: a plurality of reservoirs comprising different nucleotide reagents; and a flow chamber for receiving the nucleotide reagents.

In another embodiment, the present invention provides a method of sequencing a polynucleotide strand contained in a well of an array having multiple wells, comprising: flowing a series of nucleotides to the array; receiving signal data relating to chemical reactions in a plurality of wells within a region of the array, wherein the plurality of wells includes the well containing the polynucleotide strand; calculating a set of averaged signal values from the signal data; and determining a region-wide signal droop rate by fitting a signal decay function to the set of averaged signal values.

In some cases, the signal decay function is an exponentially decaying function. In some cases, the method further comprises performing a base calling analysis of the signal data from the well containing the polynucleotide strand using the region-wide signal droop rate. In some cases, the method further comprises: applying the region-wide signal droop rate to the signal decay function; and fitting the signal decay function to the signal data from the well containing the polynucleotide strand to obtain an individual-read signal droop rate. In some cases, the method further comprises performing a base calling analysis of the signal data from the well containing the polynucleotide strand using the individual-read signal droop rate.

In some cases, the method further comprises: constructing a phase-state model to simulate a population of the polynucleotide strands for a set of nucleotide flows, wherein the model includes one or more parameters for incomplete extension rate, carry forward rate, or both; using the phase-state model and the signal droop rate to calculate predicted signals resulting from the set of flows; and performing base calls by comparing the predicted signals to the signal data.

In another embodiment, the present invention provides a non-transitory machine-readable storage medium comprising instructions which, when executed by a processor, cause the processor to perform the above-described methods. In some cases, the processor is caused to: receive signal data relating to chemical reactions in a plurality of wells within a region of a well array that is subjected to the flow of a series of nucleotides; calculate a set of averaged signal values from the signal data; and determine a region-wide signal droop rate by fitting a signal decay function to the set of averaged signal values.

In another embodiment, the present invention provides an apparatus comprising: a machine-readable memory; and a processor configured to execute machine-readable instructions, said instructions which when executed cause the apparatus to perform the above-described methods. In some cases, the apparatus is programmed to: receive signal data relating to chemical reactions in a plurality of wells within a region of a well array that is subjected to the flow of a series of nucleotides; calculate a set of averaged signal values from the signal data; and determine a region-wide signal droop rate by fitting a signal decay function to the set of averaged signal values. In some cases, the apparatus further comprises: a plurality of reservoirs comprising different nucleotide reagents; and a flow chamber for receiving the nucleotide reagents.

FIG. 1 illustrates a simplified example of a base calling process in sequencing-by-synthesis. Specifically, FIG. 1 shows a DNA fragment inside a reaction well as it is undergoing sequencing reactions. Sequencing operations produce signal data used to make base calls of the sequence. There is a template strand 20 that is paired with a growing complementary strand 22. In the left panel, an A nucleotide is added to the reaction well, resulting in a single-base incorporation event which generates a single hydrogen ion. In the right panel, T nucleotides are added to the reaction well, resulting in a two-base incorporation event which generates two hydrogen ions. The signals used for making base calls can be represented in an ionogram, which is a graphical representation of the signals received from the sequencing operations after the raw data signals have been processed. The signal produced by the hydrogen ions are shown as peaks 26 in the ionograms.

FIG. 2 shows an example ionogram from which base calls can be made. In this example, the x-axis shows the nucleotide species that is flowed and the number of nucleotide bases incorporated can be inferred by rounding to the nearest integer shown in the y-axis.

Generally, the raw signals received from the sequencing reactions are processed before they are used to make base calls. This signal processing may include making adjustments to the signals to correct for signal droop and phasing effects (as will be explained in more detail below). To demonstrate the effect of phasing correction, FIG. 3A shows an ionogram for a sequencing read prior to signal correction for phasing effects. FIG. 3B shows an ionogram for the same sequencing read after signal correction for phasing effects. More accurate estimates of the phasing effects and/or signal droop can improve the ability to make accurate base calls.

Embodiments of the present invention address the problem of improving the accuracy of base calling in sequencing-by-synthesis of a template polynucleotide strand. The sequencing operations produce measured signal data and this measured signal data is used to make base calls. The signal data used to make the base calls may be taken from any suitable point in the acquisition or processing of the signals received from the sequencing reactions. For example, the signal data may be obtained from the raw signals resulting from the chemical reactions or from a signal obtained after the raw signal has undergone further processing, such as background filtering, normalization, etc. The base calls may be made by analyzing any suitable characteristic of the signal data, such as the signal amplitude (e.g., signal intensity).

Ideally, each extension reaction associated with the population of template polynucleotide strands are performing the same incorporation step at the same sequence position in each flow cycle, which can generally be referred to as being “in phase” or in “phasic synchrony” with each other. However, it has been observed that some fraction of template strands in each population may lose or fall out of phasic synchronism with the majority of the template strands in the population. That is, the incorporation events associated with a certain fraction of template strands may either get ahead of or fall behind other template strands in the sequencing run. Such phase loss effects are described in Ronaghi, GENOME RESEARCH, 11:3-11 (2001); Leamon et al., CHEMICAL REVIEWS, 107:3367-3376 (2007); Chen et al., International Patent Publication WO 2007/098049. Such phasing effects can introduce noise into the signal, thus hindering the ability to make accurate base calls from the signals.

As mentioned above, one cause of phase synchrony loss is the failure of a sequencing reaction to incorporate one or more nucleotide species on a template strand for a given flow cycle may result in that template strand being behind the main template population in sequence position. This effect is referred to as an “incomplete extension” error. Another cause of phase synchrony loss is the improper incorporation of one or more nucleotide species on a template strand may result in that template strand being ahead of the main population in sequence position. This is referred to as a “carry forward” error. Carry forward errors may result from the misincorporation of a nucleotide species, or in certain instances, where there is incomplete removal of a previous nucleotide species in a reaction well (e.g. incomplete washing of the reaction well). Thus, as a result of these phasing effects, at a given flow cycle, the population of template strands may be a mixture of strands in different phase-states. As noted above, the out-of-phase template strands can introduce noise into the signal, making accurate base calling more difficult.

According to exemplary embodiments of the present invention, more accurate estimates of signal correction parameters that can be applied to the signal analysis used for base calling are provided. As used herein, “signal correction parameter” means any parameter that can be applied to the analysis of the measured signal data to account for effects that increase noise, blur the signal, or otherwise affect the signal in such a way to impair the ability to make accurate base calls. Examples of signal correction parameters include phasing effect parameters, such as the incomplete extension rate and the carry forward rate, and signal droop (sometimes also referred to as signal decay).

According to exemplary embodiments of the present invention, these estimates of signal correction parameter(s) may be obtained by using a statistical model of the sequencing process. The model may be constructed in any suitable fashion to mathematically describe the sequencing process. In some cases, the model is a phase-state model that simulates a population of template strands as it undergoes the sequencing process and becomes divided into different phase-states as the sequencing-by-synthesis progresses. The number of template strands in each phase-state may be calculated using one or more phasing effect parameters, such as the incomplete extension rate and/or the carry forward rate.

Each phase-state may represent a template strand having a different number of nucleotide bases incorporated. In other words, template strands in different phase-states have a different number of nucleotide bases incorporated. In some cases, homopolymer incorporations (more than a single incorporation) on homopolymer stretches of the template may be considered to occupy the same state. That is, an n-mer base incorporation would take up one phase-state regardless of whether its length is 1-mer, 2-mer, 3-mer, etc. The term “n-mer” refers to the number of contiguous identical complementary bases that are incorporated into the complementary strand on the template strand. If the next base in the template strand is not complementary to the flowed nucleotide, generally no incorporation occurs and the resulting output signal is sometimes referred to as a “0-mer” output signal.

According to exemplary embodiments of the present invention, the different phase-states of the template population are represented in a phasing matrix, in which the flow cycles are represented on one axis and the different phase-states of the template strands are represented on the other axis. Each entry in this matrix contains a value that is related to the number of template strands that occupy that phase-state at that particular flow cycle. Thus, the phasing matrix gives the portion of the population of clonal template strands that occupy each phase-state, which can be calculated using the incomplete extension rate and/or the carry forward rate.

Since the phase-states of the template will depend on the sequence of the template strand and the nucleotide flow ordering, the model may be constructed based on a given nucleotide flow ordering and at least some sequence information about the template strand. Although the sequencing run may have obtained the signal data for the full read of the template strand, this sequence information used by the model may or may not be the complete sequence of the template strand. For example, this sequence information may only represent a portion of the template strand. In some cases, the sequence information may already be known as the true sequence. In some cases, the sequence may not be known beforehand and a preliminary estimate of the sequence is obtained. For example, a preliminary estimate of the sequence can be obtained by establishing preliminary, naive thresholds for the measured signal data to make preliminary base calls and generate the preliminary sequence information. For example, a measured signal value (after normalization to a key sequence) in the range of 0.5 to 1.5 may be called as a 1-mer, a range of greater than 1.5 up to 2.5 may be called as a 2-mer, and so on. As explained below, this preliminary sequence information may be revised as the signal correction parameter(s) are adjusted to provide more accurate signal analysis.

An example of how a phase-state matrix may be constructed is shown in Table 1A below, which shows a phase matrix that simulates a template population of 100,000 identical strands of DNA. The phase matrix is constructed under the following conditions:

-   -   Incomplete extension rate: 1%     -   Carry forward rate: 1%     -   Signal droop rate: 0%     -   Preliminary DNA sequence information: AGTC (nucleotides         incorporate with the complement TCAG)     -   Flow order: T-A-C-G-T-A

Each entry in the matrix is the number of template strands out of the 100,000 that are in a particular phase-state (the blank entries are zero) after the nucleotide flow occurs. The top row of the matrix shows the flow number and the leftmost column shows the phase-state condition of the template strands. Prior to any nucleotide flows taking place (flow 0 or initial state), each of the 100,000 strands is in phase-state 0. As the flow cycles progress, individual template strands in the population move to different phase-states. The number of strands that appropriately proceed to a subsequent phase-state (e.g. next phase state), or inappropriately fail to proceed to a subsequent phase-state (e.g. incomplete extension error), or inappropriately proceed to a subsequent phase-state (e.g. carry forward error) will depend upon various factors, including the rates of incomplete extension and carry forward errors. In this example, the incomplete extension and the carry forward rate are both set at an initial estimate of 1%.

Table 1B below shows an alternate construction of this phase matrix for the same population of template strands, but taking into account the effects of signal droop. In this case, a signal droop rate of 1% per phase-state change is applied to the phase-state model by reducing each phase-state transition by 1% to represent the effect of signal droop.

TABLE 1A Flow 1 Flow 2 Flow 3 Flow 4 Flow 5 Flow 6 Initial (T) (A) (C) (G) (T) (A) State 100,000 1,000 1,000 1,000 1,000 10 10 0 State 99,000 99,000 990 990 1,980 1,980 1 State 97,040 97,040 97,040 970 2 State 970 10 10 96,080 3 State 960 960 960 4

TABLE 1B (with 1% signal droop rate). Flow 1 Flow 2 Flow 3 Flow 4 Flow 5 Flow 6 Initial (T) (A) (C) (G) (T) (A) State 100,000 1,000 1,000 1,000 1,000 10 10 0 State 98,010 98,010 980 980 1,960 1,960 1 State 95,109 95,109 95,109 951 2 State 941 9 9 93,225 3 State 923 923 923 4

The phase matrix in Table 1B is explained in more detail as follows. In this example, prior to any nucleotide flows taking place (flow 0 or initial state), each of the 100,000 strands is in state 0. In the first flow, the nucleotide species T is flowed. Since this nucleotide species is complementary to the first base in the AGTC template sequence, this should ideally result in the incorporation of the T nucleotide species in all 100,000 strands, which would all move from state 0 to state 1. However, because of the 1% incomplete extension rate, some of the strands remain in the prior state. Also, because of the 1% droop rate, some of the strands are removed from the population to represent this loss. In this case, 99,000 strands advance. However, some of those advancing strands will experience droop (polymerase loss for example) and fail to further incorporate; thus 98,010 strands are now present in state 1, and 1,000 strands (i.e. 1% of the population) remains in state 0.

In the second flow, the nucleotide species A is flowed. Since this nucleotide species is not complementary to the second base in the AGTC template sequence, there is no incorporation event and none of the templates move to phase-state 2. In the third flow, the nucleotide species C is flowed. Since this nucleotide species is complementary to the second base in the AGTC template sequence, this should ideally result in the incorporation of the C nucleotide species in all 98,010 strands in phase-state 1. However, after accounting for the 1% incomplete extension rate and the 1% droop rate, 97,040 strands advance and 970 strands remain in phase-state 1. Moreover, because of the 1% carry forward rate, some of the template strands that advanced into state 2 further advance to phase-state 3. The population is reduced with each advance to represent the loss from signal droop (on a per phase-state change basis). As shown in Table 1B, this process is repeated through flows 4-6 in a cascading fashion to build the phase matrix.

In the fourth flow, the nucleotide species G is flowed. Since this nucleotide species is not complementary to the third base in the AGTC template sequence, there is no incorporation in the main population in phase-state 2. Note however that those template strands that carried forward to phase-state 3 during flow #3 continue to carry forward to phase-state 4 in flow #4 in the matrix shown in Table 1B. In the fifth flow, the nucleotide species T is flowed. Since this nucleotide species is not complementary to the third base in the AGTC template sequence, there is no incorporation in the main population in phase-state 2. Note however that 99% of those template strands that carried forward to phase-state 3 during flow #3 and phase-state 4 during flow #4 continues to carry forward to phase-state 5 in flow #5. Also, the template strands that were in phase-state 0 (due to incomplete extension in flow #1 of the T nucleotide species) have now moved to phase-state 1 with this second flow of the T nucleotide species.

In the sixth flow, the nucleotide species A is flowed. Since this nucleotide is complementary to the third base in the AGTC template sequence, the effect of this incorporation event is shown in Table 1B. In particular, 99% of the template strands that were in phase-state 2 incorporate properly and advance to phase-state 3. Also note that 99% of those strands that carried forward to phase-state 3 during flow #3, phase-state 4 during flow #4, and phase-state 5 during flow #5 continues to carry forward to phase-state 6 in flow #6. Also, with each group of advancing strands, some portion of the population is removed to account for signal droop.

Table 1A shows a similar progression of phase-states, but without accounting for signal droop. In this situation, any signal droop can be factored separately into the calculation for the predicted signals. Since the phase-state model provides a simulation of the phase-state for each strand within the population of templates, this simulation can be used to predict the signals that would be expected if there were nucleotide incorporations, or no incorporation in the next flow. This can be done for a series including 0-mer, 1-mer, 2-mer, and so on, in order to generate a list of predicted signals. These model-predicted signals can be compared against the actually measured signals to fit the model to the measured signal data. Table 2A below shows a comparison of the model-predicted signals and the actual measured signals for the above-described example in Table 1A.

The predicted signal can be calculated in any suitable manner based on the relationship between the signal intensity and the number of nucleotides incorporated. In this example, the predicted signal is proportional to the total number of nucleotide incorporations that occur in that flow, normalized to 100,000 total nucleotide incorporations producing a normalized signal of 1.0000. For example, in Table 1A, there are 97,040 nucleotide incorporations that occur in flow #3 when the template population in state 1 moves to state 2. Additionally, there are 1,940 nucleotide incorporations that occur in flow #3 when 970 of the template strands in state 1 move to state 2, and then move further from state 2 to state 3 because of the 1% carry forward rate. Thus, there are a total of 98,980 nucleotide incorporations that occur in flow #3. When divided by 100,000 for normalization, the predicted signal at flow #3 is calculated to be 0.9898, as shown in Table 2A. The predicted signals in Table 2A can be further adjusted to account for any signal droop. For example, each predicted signal value can be reduced by the amount of signal droop. The predicted signals in Table 2B are generated from the phase matrix of Table 1B, which has already accounted for the signal droop rate. Thus, the predicted signals in Table 2B represent signals that would be expected with the effect of signal droop.

TABLE 2A Flow 1 Flow 2 Flow 3 Flow 4 Flow 5 Flow 6 (T) (A) (C) (G) (T) (A) Predicted 0.9900 0.0000 0.9898 0.0096 0.0099 0.9607 Measured 1.0200 0.0035 1.0049 0.0170 0.0237 0.9804

TABLE 2B (with a 1% signal droop rate). Flow 1 Flow 2 Flow 3 Flow 4 Flow 5 Flow 6 (T) (A) (C) (G) (T) (A) Predicted 0.9900 0.0000 0.9798 0.0093 0.0099 0.9416 Measured 1.0200 0.0035 1.0049 0.0170 0.0237 0.9804

Any suitable fitting technique can be used to fit the model to the measured signal data, such as maximum likelihood, regression analysis, and Bayesian techniques. In some cases, a fitting metric that measures the fit between the model-predicted signals and the measured signal data can be used for the fitting process. For example, the fitting metric may be the sum of the absolute differences between the measured signals and the model-predicted signals (sum of residual errors) at each flow divided by the number of flows. In another example, the fitting metric may be the square root of the sum of the squared residuals (measured signal−predicted signal). In other examples, weighted coefficients may be applied to the residuals or other properties of the predictions themselves may be used in the calculation of the fitting metric.

The fitting metric can encompass all the nucleotide flows or less than all the nucleotide flows. In some cases, the fitting metric is applied to all the flows that have been modeled or all the flows that encompass the sequence information. In some cases, only a certain range of flows may be considered in the fitting metric, for example, flows numbers 12-60 or flows numbers 12-40. In some cases, certain flows may be excluded. For example, the flows through a key sequence and/or barcode sequence in the template strand may be ignored in the fitting metric (e.g. the first 8 or 12 flows may be excluded). In another example, flows that result in larger homopolymers (e.g. 3-mers or longer) may be ignored. In another example, only flows that result in non-incorporation or single nucleotide incorporations may be considered in the fitting metric.

The model fitting may involve an iterative process of varying one or more signal correction parameters of the phase-state model to improve the fit (e.g. obtaining the best fit by minimizing the residual error sums) between the model-predicted signals and the measured signal data. Any suitable optimization algorithm may be used in the process of finding model parameter(s) that produce predicted signal values having improved fit with the measured signal data, which may include a gradient-descent type algorithm that varies the signal correction parameter(s) of the model until they converge on the solution with the minimal residual error.

This fitting step may result in an improved estimate of the signal correction parameter(s). In a microwell array, there are many wells that contain template strands that undergo simultaneous or parallel sequencing. As such, there may be multiple reads available for obtaining improved estimates of the signal correction parameter(s). In other words, when sequencing on a microwell array, there may be multiple sets of measured signal data, each representing a read from one of the multiple wells of the microwell array. In such instances, the above-described process may be applied to other sets of measured signal data to obtain additional estimates for signal correction parameter(s).

In some cases, the fitted signal correction parameter(s) may be obtained for each individual well read and be applied to the measured signal data for that individual well only. Alternatively, in some cases, regional estimates may be taken where the fitted signal correction parameter(s) from multiple individual wells in a selected region of the microwell array (e.g. a region size of 100×100 wells) are taken and statistically analyzed to obtain a region-wide estimate of the parameter(s). The signal data from the group of wells may be subject to any suitable statistical analysis to obtain a single value as a region-wide estimate that quantitatively summarizes the collection of signal data, including calculating an average, a weighted average, some function of the average, a mean, or a mode of the signal data.

This region-wide estimate of the parameter(s) may also be further refined. In some cases, sequence reads that produce poorly fitting measured signal data may be excluded from the calculation of the region-wide estimate. For example, reads with high residual errors in the fitting (e.g. exceeding a maximum residual error threshold) or those that are poorly fitting (e.g. exceeding a boundary condition) may be excluded from the calculation of the region-wide estimate.

In some cases, the estimated signal correction parameter(s) may be subject to further statistical refinement. For example, a truncated mean of the multiple estimates may be taken, in which any value outside of a certain threshold range around the calculated mean (e.g. outside a 60% window around the calculated mean) is excluded from the group and the average is recalculated. In some cases, the quality of the individual well reads may be considered in whether to include the individual read in the region-wide estimate. For example, lower quality reads may be excluded, such as mixed reads (i.e. reads from wells containing more than one kind of template strand).

Another problem that can hinder accurate base calling in sequencing-by-synthesis is the decay of the signal (often referred to as signal droop). There are a number of possible reasons for decay of the signal, including loss of DNA polymerase activity, template strands that are unable to incorporate any more nucleotides, washing out of the polymerase or template strands, loss in sensitivity of the well sensors, etc. The signal droop rate can be given in any suitable unit basis, including per flow, per nucleotide incorporation, or per phase-state change.

In another aspect, the present teachings provide a method for obtaining a more accurate estimate of signal droop rate. In some cases, the signal droop rate may be modeled as a decaying function, such as a linearly decaying function or an exponentially decaying function. An example of an exponentially decaying function that can be used for estimating signal droop is expressed by the following equation: S_(e)(N)=S_(b)(1−Dpf)^(N)

where S_(e) is the expected signal at a given flow N; S_(b) is the baseline signal; Dpf is the signal droop per flow; and N is the flow number. This equation can be solved for signal droop rate as a per-flow decay. To obtain an estimate of signal droop rate on a per-nucleotide incorporation basis, the per-flow droop rate can be divided by the baseline signal.

By varying the signal droop rate parameter, the signal decay function may be fitted to the measured signal data to obtain a more accurate estimate of the signal droop rate. In some cases, the signal decay function may be fitted to an average of the measured signal data taken from multiple wells in a region of the microwell array (e.g. a region size of 100×100 wells). On a microwell array where different wells may contain different fragments of a genome, only some of the fragments will undergo an incorporation for a given flow. But because the different genomic fragments may be considered a random sampling of nucleotide bases at a given flow, regional averages may be useful for fitting the signal decay function.

The fitting of the signal droop rate may encompass all the nucleotide flows or less than all the nucleotide flows. In some cases, only a certain range of flows may be considered in the fitting. In some cases, certain flows may be excluded. For example, the flows through a key sequence and/or barcode sequence in the template strand may be ignored in the fitting because these sequences may not represent diverse or random sequences.

For example, in a typical genome, the probability that the next flow will be a complementary match to the next base in the template stand is 0.333. Moreover, the probability that the next base is a 1-mer, 2-mer, 3-mer, etc can also be estimated. Given these two estimates, the average signal measured in a given nucleotide flow for a random population of sequencing templates can then be estimated by the following equation: average signal=P1×Pb×S1+P2×Pb×S2+P3×Pb×S3+ . . . ; where S1, S2, S3 are the strengths of the signals received for 1-mer, 2-mer, 3-mer etc. incorporations, respectively; P1, P2, P3 are the probabilities that for the given incorporation, that incorporation is a 1-mer, 2-mer, 3-mer, etc. respectively; Pb is the probability that the next nucleotide flow matches the next base in the template (i.e. the probability that an incorporation event will occur).

This equation can be rewritten as follows: average signal=Pb×(P1×S1+P2×S2+P3×S3+ . . . ). Given a repeated flow ordering of T-A-C-G, for example, Pb must be ⅓ because during the prior nucleotide flows, one of the flowed nucleotides would have incorporated and a repeated flow of the same nucleotide would not cause an incorporation. Thus, only 3 nucleotide species remain as possible incorporating nucleotides. Therefore, the average signal measured in a nucleotide flow is the sum of the signal received for a 1-mer incorporation (S1) times the probability that the next base will incorporate as a 1-mer (P1), plus the probability that the next base will incorporate as a 2-mer (P2) times the signal generated for that 2-mer (S2; signal is twice a 1-mer when a 2-mer incorporates), plus the probability that the next base will incorporate as a 3-mer times the signal generated for that 3-mer, etc.

Using the foregoing calculations, the expected average of the measured signals for any given flow over many random sequences may be about 0.6. This estimated average of 0.6 is not critical, but this example provides a basis for illustrating the concept of signal droop. Here, 0.6 is the baseline signal (denoted as S_(b) above) expected before any signal decay has occurred. Tracing the region-wide average signal over successive flows produces a curve that depicts the decay of the signal. Fitting the signal decay function to this curve (by varying the signal droop rate parameter) can produce an improved estimate of the signal droop rate.

Any suitable statistical technique or optimization algorithm may be used to fit the signal decay function to the averaged signal values. The signal droop rate can be provided as a per-region estimate or a per-read estimate (i.e. the read from an individual well). Although the per-region estimates can be obtained faster and more easily, a per-read estimate can be more accurate. As such, in some cases, a per-region estimate can be used as an initial guess of the actual per-read droop. Then, this initial guess can be used in the fitting to obtain a per-read estimate. By starting with a good initial guess, the fitting can converge to an accurate solution more rapidly with less susceptibility to converging on local minima.

After the fitted signal correction parameter(s) are obtained, they can be applied to the signal analysis process that results in base calling. Where a region-wide estimate of the signal correction parameter(s) is used, that region-wide estimate may be applied to the reads (e.g. all the reads) in that particular region. In some cases, the signal correction parameter(s) can be applied to the measured signal data to produce corrected signal data. The corrected signal data can then be used for making base calls. In such cases, any of various types of signal correction techniques may be used for the base calling. For example, the signal correction may be performed using the technique described in H. Eltoukhy & A. El Gamal, “Modeling and Base-Calling for DNA Sequencing-By-Synthesis,” at 2006 IEEE International Conference on Acoustics, Speech, and Signal Processing (May 2006). In another example, the signal correction may involve the use of the technique described in U.S. Patent Application Publication No. 2010/0192032 (Chen et al.; 454 Life Sciences Corp.), which is incorporated by reference herein.

In some cases, the signal analysis for base calling may involve using the above-described phase-state model in a predictive mode where, for a given flow, one or more predicted signal values are generated. Since they are calculated from the phase-state model, the predicted signal value(s) are the signal values that would be expected after correcting for phasing effects and/or signal droop. Having these predicted signal values at different flows, base calls can be made by comparing the actually measured signal at a particular flow against the predicted signal value at the same flow.

In some embodiments, multiple predicted signal values can be generated for each flow being considered, each of the predicted signal values being associated with a potential incorporation event (including non-incorporation) that could occur at that flow. For example, Table 3 below shows a list of multiple predicted signal values across several flows over the preliminary sequence information AGTC described above with respect to Table 1. At each flow, there are predicted signal values for a non-incorporation event and the predicted signal values for n-mer incorporations (up to 7-mers). Since the list of predicted signal values encompass multiple possible incorporation events, these predictions can be applied to unknown sequences to make base calls. In this example, the set of predictions is up to 7-mer incorporations per nucleotide flow, but this is for illustration purposes only and does not limit any particular aspect of the present teachings in general. In practice, predicted values can be generated for any number of polymer incorporations (e.g. predictions for 8-mer incorporations, 9-mer incorporations, and so on). The bolded entries in Table 3 indicate the best match against the measured signal data and indicate the incorporation event for the given flow.

TABLE 3 0-mer 1-mer 2-mer 3-mer 4-mer 5-mer 6-mer 7-mer Flow 1 (T) 0.0000 0.9900 1.9800 2.9700 3.9600 4.9500 5.9400 6.9300 Flow 2 (A) 0.0000 0.9801 1.9602 2.9403 3.9204 4.9005 5.8806 6.8607 Flow 3 (C) 0.0097 0.9898 1.9699 2.9500 3.9301 4.9102 5.8903 6.8704 Flow 4 (G) 0.0096 0.9703 1.9310 2.8917 3.8524 4.8131 5.7738 6.7344 Flow 5 (T) 0.0099 0.9609 1.9119 2.8629 3.8139 4.7649 5.7158 6.6668 Flow 6 (A) 0.0000 0.9607 1.9214 2.8821 3.8428 4.8035 5.7641 6.7248

In this example, the signal values are assumed to be generally proportional to the amount of nucleotides incorporated. Thus, for each flow, each predicted signal value associated with a multiple nucleotide incorporation (greater than 1) is calculated as the predicted signal from a single nucleotide incorporation multiplied by the number of incorporations. Thus, each predicted signal value for multiple n-mer (greater than 1) incorporations is calculated using the following equation: P(n-mer)=B+(D×n-mer), where P(n-mer) is the predicted signal for the homopolymer n-mer, B is the baseline 0-mer signal value, and D is the signal value for a single nucleotide incorporation (which is the difference in the 1-mer and 0-mer signals, not the absolute predicted 1-mer signal). For example, at flow number 3, P(2-mer)=0.0097+(0.9898−0.0097)×2=1.9699. Given a list of predicted signal values for each potential incorporation event, the actually measured signal at that flow can be compared against the predicted signal values and the incorporation event associated with the predicted signal value that most closely matches the measured signal value is made as the base call for that flow cycle.

As a result of this process, more accurate sequence information about the template strand may be obtained. For example, if the preliminary sequence information was based on naïve thresholding, this preliminary sequence can be revised based on analysis of the same measured signal data with improved signal correction parameter(s). Thus, the sequence information may be revised, updated, and/or extended over a greater length with successive iterations of this algorithm.

In some embodiments, the above-described algorithm can be repeated to obtain further, more accurate estimates of the signal correction parameter(s). In some cases, subsequent passes of the iterative process may use a larger set of flows encompassing more of the template strand. For example, if the preliminary sequence information was obtained for only a portion of the template polynucleotide strand, then this algorithm may be repeated for a slightly larger portion of the template strand (and its associated nucleotide flows) to obtain further estimates of the signal correction parameter(s). Also, this iterative process can continue until a sufficient level of confidence in the accuracy of the base calls is achieved. For example, this iterative fitting and re-fitting process over incrementally larger sets of flows can continue until about 100 flows. After this has been repeated for the desired number of flows (e.g. 100 flows), the final signal correction parameter(s) can then used to solve all flows in the manner described above.

For example, the phase-state model may be applied multiple times to the measured signal data. In the first pass, the measured signal data is compared against naïve thresholds to generate an initial guess for the sequence of a portion of the template strand (e.g. the first N bases). In the first pass, the phase-state model may be used with only the effect of incomplete extensions being considered. In subsequent passes, the phase-state model may consider both the effects of incomplete extensions and carry forward effects.

A flow chart illustration of an embodiment of the present teachings is shown in FIG. 4. The measured signal data, the preliminary sequence information (e.g. obtained by base calling using naïve thresholds), the nucleotide flow ordering, and one or more signal correction parameters (e.g. carry forward, incomplete extension, and/or signal droop) are used as inputs to the phase-state model. From the template populations simulated in the model, the predicted signal values over the preliminary sequence information are calculated. The predicted signal values are compared against the measured signal data. In an iterative process, the signal correction parameter(s) are varied and applied as input to the phase-state model until the desired level of optimized fit (e.g. best fit) between the predicted signal values and the measured signal data is achieved. Using the fitted signal correction parameter(s), the measured signal data is analyzed to make base calls that revise, update, and/or extended the sequence information. This new sequence information can be applied to the phase-state model for further iterative passes of the algorithm.

In some cases, the signal droop rate is determined independently from the phasing parameter(s), and in some cases, the same data set may be used in the independent determinations. As explained above, in some cases, the signal droop rate and the phasing parameter(s) may be applied separately to calculate the predicted signals. For example, in the embodiment shown in FIG. 5, a signal droop model is used to calculate a signal droop rate and a phase-state model is used to calculate the rates of incomplete extension (IE) and carry forward (CF). Both models are applied to the signal data and may use the same or different portions of the signal data. Outputs from both models may be used to generate the set of predicted signals. Alternatively, as explained above, the phase-state model itself can be adjusted for the signal droop rate (e.g. by applying reductions on a per phase-state change basis). In this case, the output of the phase-state model alone may be used to calculate the predicted signals.

The present teachings may use any of various techniques for detecting the nucleotide incorporation(s). For example, some sequencing-by-synthesis techniques operate by the detection of pyrophosphate (PPi) released by the incorporation reaction (see e.g., U.S. Pat. Nos. 6,210,891; 6,258,568; and 6,828,100). In another example, some sequencing-by-synthesis techniques detect labels associated with the nucleotides, such as mass tags, fluorescent, and/or chemiluminescent labels. Where detectable labels are used, an inactivation step may be included in the workflow (e.g. by chemical cleavage or photobleaching) prior to the next cycle of synthesis and detection. The present teachings may be particular useful for sequencing methods that operate by single-nucleotide addition, in which the precursor nucleotides are repeatedly added individually to the reaction in series according to a predetermined ordering. Examples of such sequencing techniques include those based on the detection of inorganic pyrophosphate or hydrogen ions produced by the incorporation reactions.

The reactions may be carried out on microwell sensor arrays, such as those described in U.S. Patent Application Publication No. 2010/0300559, No. 2010/0197507, No. 2010/0301398, No. 2010/0300895, No. 2010/0137143, and No. 2009/0026082, and U.S. Pat. No. 7,575,865, which are incorporated by reference herein. The microwell sensor array has multiple wells for carrying out the sequencing reactions. In some cases, the sensor array is a chemFET sensor array. In some cases, the chemFET sensors of the sensor array detects hydrogen ions. In some cases, flowing of the reagent(s) onto the sensor array causes chemical reactions that release hydrogen ions. In some cases, the amplitude of the signals from the chemFET sensors is related to the amount of hydrogen ions detected. In some cases, the sensor array is a light-sensing array. In some cases, flowing of the reagent(s) onto the sensor array causes chemical reactions that release inorganic pyrophosphate, which causes the emission of light via an enzyme cascade initiated by the inorganic pyrophosphate.

In certain embodiments, the present teachings may use a pH-based method of detecting nucleotide incorporation(s). Such an approach may measure the amount of hydrogen ions released from the polymerase-catalyzed incorporation reactions. In pH-based methods for DNA sequencing, base incorporations can be determined by measuring the hydrogen ions that are generated. Additional details of pH-based sequence detection systems and methods can be found in commonly-assigned U.S. Patent Application Publication No. 2009/0127589 and No. 2009/0026082, which are incorporated by reference.

In pH-based detection methods, the production of hydrogen ions may be monotonically related to the number of contiguous complementary bases in the template strands (as well as the total number of template strands with primer and polymerase that participate in an extension reaction). Thus, when there is a number of contiguous identical complementary bases in the template (i.e. a homopolymer region), the number of hydrogen ions generated is generally proportional to the number of contiguous identical complementary bases. The corresponding output signals may sometimes be referred to as “1-mer”, “2-mer”, “3-mer” output signals, and so on, based on the expected number of repeating bases. Where the next base in the template is not complementary to the flowed nucleotide, generally no incorporation occurs and there is no substantial release of hydrogen ions (in which case, the output signal is sometimes referred to as a “0-mer” output signal).

In each wash step of the cycle, a wash solution (typically having a predetermined pH) is used to remove residual nucleotide of the previous step in order to prevent misincorporations in later cycles. Usually, the four different kinds of nucleotides (e.g. dATP, dCTP, dGTP, and dTTP) are flowed sequentially to the reaction chambers, so that each reaction is exposed to one of the four different nucleotides for a given flow, with the exposure, incorporation, and detection steps being followed by a wash step. An example of this process is illustrated in FIG. 6, which shows a template polynucleotide strand 682 attached to a particle 680. Primer 684 is annealed to template strand 682 at its primer binding site 681. A DNA polymerase 686 is operably bound to the template-primer duplex. Template strand 682 has the sequence 685, which is awaiting complementary base incorporation. Upon the flow of the nucleotide (shown as dATP), polymerase 686 incorporates a nucleotide since “T” is the next nucleotide in template strand 682 (because the “T” base is complementary to the flowed dATP nucleotide). Wash step 690 follows, after which the next nucleotide (dCTP) is flowed 692. Optionally, after each step of flowing a nucleotide, the reaction chambers may be treated with a nucleotide-destroying agent (such as apyrase) to eliminate any residual nucleotides remaining in the chamber, which can cause spurious extensions in subsequent cycles.

The present teachings also provide an apparatus for sequencing polynucleotide strands according to the method of the present teachings. A particular example of an apparatus of the present teachings is shown in FIG. 7. The apparatus of FIG. 7 is configured for pH-based sequencing and includes multiple reservoirs for containing nucleotide reagents 1 through K (114). These reagents contain the nucleotides to be flowed for the sequencing process. The reagents 114 are flowed through fluid passages 130 and through a valve block 116 that controls the flow of the reagents to flow chamber 105 (also referred to herein as a reaction chamber) via fluid passage 109. The apparatus includes a reservoir 110 for containing a wash solution that is used to wash away the nucleotide reagent of the previous step. Reagents are discarded through waste passage 104 into a waste container 106 after exiting the flow chamber 105.

The apparatus also includes a fluidics controller 118, which may programmed to control the flow from the multiple reagent reservoirs to the flow chamber according to a predetermined ordering that comprises an alternate flow ordering, as described above. For this purpose, fluidics controller 118 may be programmed to cause the flow of reagents 114 from the reagents reservoir and operate the valves 112 and 116. The fluidics controller may use any conventional instrument control software, such as LabView (National Instruments, Austin, Tex.). The reagents may be driven through the fluid pathways 130, valves, and flow cell by any conventional mechanism such as pumps or gas pressure.

The apparatus also has a valve 112 for controlling the flow of wash solution into passage 109. When valve 112 is closed, the flow of wash solution is stopped, but there is still uninterrupted fluid and electrical communication between reference electrode 108, passage 109, and sensor array 100. Some of the reagent flowing through passage 109 may diffuse into passage 111, but the distance between reference electrode 108 and the junction between passages 109 and 111 is selected so that little or no amount of the reagents flowing in common passage 109 reach reference electrode 108. This configuration has the advantage of ensuring that reference electrode 108 is in contact with only a single fluid or reagent throughout an entire multi-step reaction process.

As shown in FIG. 7, flow chamber 105 is loaded with a flow cell that includes an inlet 102, an outlet 103, and a microwell array 107 which is operationally associated with a sensor array 100 that measures physical and/or chemical parameters in the microwells that provide information about the status of a reaction taking place therein; or in the case of empty wells, information about the physical and/or chemical environment in the flow cell. Each microwell may have a sensor for detecting an analyte or reaction property of interest. In this particular embodiment, the microwell array is integrated with the sensor array as a single chip. A flow cell can have a variety of designs for controlling the path and flow rate of reagents over the microwell array. This particular apparatus has an array controller 126 which receives information from sensor array 100 and reference electrode 108 via communication line 126. A user interface 128 provides an interface through which a user may interact with the apparatus.

An apparatus may be used to perform the above-described methods of the present teachings. The apparatus may be a computer that includes various components such as processor(s) and memory. An example of an apparatus of the present teachings is shown in FIG. 8. In some embodiments, the apparatus 600 may include one or more processors 604 and machine-readable memory 606. In some embodiments, the apparatus may include a display 608. In some embodiments, the apparatus may include a reader board 610 which is coupled to a sensor array 618. The reader board 610 may include various components used in signal processing, including analog-to-digital converters. In some embodiments the apparatus may be part of the sequencing apparatus. In other embodiments, the apparatus may be separate from the sequencing apparatus; in some embodiments the apparatus may be coupled to the sequencing apparatus.

In various embodiments, a polynucleotide may be represented by a sequence of letters (upper or lower case), such as “ATGCCTG,” and it will be understood that the nucleotides are in 5′→>3′ order from left to right and that “A” denotes deoxyadenosine, “C” denotes deoxycytidine, “G” denotes deoxyguanosine, and “T” denotes thymidine, “I” denotes deoxyinosine, “U” denotes uridine, unless otherwise indicated or obvious from context.

Polynucleotides may comprise the four natural nucleosides (e.g. deoxyadenosine, deoxycytidine, deoxyguanosine, deoxythymidine for DNA or their ribose counterparts for RNA) linked by phosphodiester linkages. However, they may also comprise non-natural nucleotide analogs, e.g. including modified bases, sugars, or internucleosidic linkages. It is clear to those skilled in the art that where an enzyme has specific oligonucleotide or polynucleotide substrate requirements for activity (e.g. single stranded DNA, RNA/DNA duplex, or the like), then selection of an appropriate composition for the oligonucleotide or polynucleotide substrates is well within the knowledge of one of ordinary skill, especially with guidance from treatises such as Sambrook et al, MOLECULAR CLONING, 2nd ed. (Cold Spring Harbor Laboratory, New York, 1989), and like references.

“Polynucleotide” refers to a linear polymer of nucleotide monomers and may be DNA or RNA. Monomers making up polynucleotides are capable of specifically binding to a natural polynucleotide by way of a regular pattern of monomer-to-monomer interactions, such as Watson-Crick type of base pairing, base stacking, Hoogsteen or reverse Hoogsteen types of base pairing, or the like. Such monomers and their internucleosidic linkages may be naturally occurring or may be analogs thereof, e.g., naturally occurring or non-naturally occurring analogs. Non-naturally occurring analogs may include PNAs, phosphorothioate internucleosidic linkages, bases containing linking groups permitting the attachment of labels, such as fluorophores, or haptens, and the like. As used herein, the term “oligonucleotide” refers to smaller polynucleotides, for example, having 5-40 monomeric units.

Various embodiments may be implemented using hardware elements, software elements, or a combination of both. Examples of hardware elements may include processors, microprocessors, circuits, circuit elements (e.g., transistors, resistors, capacitors, inductors, and so forth), integrated circuits, application specific integrated circuits (ASIC), programmable logic devices (PLD), digital signal processors (DSP), field programmable gate array (FPGA), logic gates, registers, semiconductor device, chips, microchips, chip sets, and so forth. Examples of software may include software components, programs, applications, computer programs, application programs, system programs, machine programs, operating system software, middleware, firmware, software modules, routines, subroutines, functions, methods, procedures, software interfaces, application program interfaces (API), instruction sets, computing code, computer code, code segments, computer code segments, words, values, symbols, or any combination thereof. Determining whether an embodiment is implemented using hardware elements and/or software elements may vary in accordance with any number of factors, such as desired computational rate, power levels, heat tolerances, processing cycle budget, input data rates, output data rates, memory resources, data bus speeds and other design or performance constraints.

Some embodiments may be implemented, for example, using a machine-readable medium or article which may store an instruction or a set of instructions that, if executed by a machine, may cause the machine to perform a method and/or operations in accordance with the embodiments. Such a machine may include, for example, any suitable processing platform, computing platform, computing device, processing device, computing system, processing system, computer, processor, or the like, and may be implemented using any suitable combination of hardware and/or software. The machine-readable medium or article may include, for example, any suitable type of memory unit, memory device, memory article, memory medium, storage device, storage article, storage medium and/or storage unit, for example, memory, removable or non-removable media, erasable or non-erasable media, writeable or re-writeable media, digital or analog media, hard disk, floppy disk, read-only memory compact disc (CD-ROM), recordable compact disc (CD-R), rewriteable compact disc (CD-RW), optical disk, magnetic media, magneto-optical media, removable memory cards or disks, various types of Digital Versatile Disc (DVD), a tape, a cassette, or the like, including any medium suitable for use in a computer. The instructions may include any suitable type of code, such as source code, compiled code, interpreted code, executable code, static code, dynamic code, encrypted code, and the like, implemented using any suitable high-level, low-level, object-oriented, visual, compiled and/or interpreted programming language.

According to other embodiments of the present teachings, any one or more features of any one or more of the above-discussed teachings and/or exemplary embodiments may be performed or implemented at least partly using a cloud computing resource.

Those skilled in the art may appreciate from the foregoing description that the present teachings may be implemented in a variety of forms, and that the various embodiments may be implemented alone or in combination. Therefore, while the embodiments of the present teachings have been described in connection with particular examples thereof, the true scope of the embodiments and/or methods of the present teachings should not be so limited since other modifications will become apparent to the skilled practitioner upon a study of the drawings, specification, and following claims. 

1. A method of sequencing a polynucleotide strand, comprising: (a) receiving signal data relating to chemical reactions resulting from a flow of nucleotides; (b) determining, using the signal data, sequence information of at least a portion of the polynucleotide strand; (c) constructing a phase-state model for a set of flows that encompasses the sequence information, wherein the model includes a signal correction parameter and wherein the model is stored in a machine-readable memory; (d) calculating, using the phase-state model, predicted signals resulting from the set of nucleotide flows; (e) comparing the predicted signals to the signal data; (f) adjusting the signal correction parameter of the phase-state model based on the comparison of the predicted signals to the signal data; and (g) storing the adjusted signal correction parameter in the memory.
 2. The method of claim 1, wherein the nucleotides are flowed onto an array having multiple wells, wherein the polynucleotide strand is contained in a first well of the array, and further comprising: obtaining signal data relating to chemical reactions in a plurality of other wells within a region around the first well; and performing steps (b) through (g) for each of the obtained signal data from the plurality of other wells to obtain multiple adjusted signal correction parameters.
 3. The method of claim 2, further comprising calculating a region-wide estimate of the signal correction parameter using the multiple adjusted signal correction parameters.
 4. The method of claim 3, wherein the comparing step comprises calculating a fitting metric that measures the fit between the predicted signals and the signal data from at least some of the plurality of wells.
 5. The method of claim 4, wherein the fitting metric measures the fit between the predicted signals and the signal data from less than all of the plurality of wells; and wherein the region-wide estimate excludes adjusted signal correction parameters from wells that produce a fitting metric exceeding a predetermined threshold.
 6. The method of claim 3, further comprising performing a base calling analysis of the signal data from multiple wells within the region using the region-wide estimate of the signal correction parameter.
 7. The method of claim 1, further comprising repeating steps (c) through (g) using the adjusted signal correction parameter.
 8. The method of claim 7, wherein use of the adjusted signal correction parameter improves the fit between the signal data and the predicted signals.
 9. The method of claim 1, wherein the phase-state model includes two or more signal correction parameters, including a carry forward rate and an incomplete extension rate.
 10. The method of claim 1, wherein the comparing step comprises calculating a fitting metric that measures a fit between the predicted signals and the signal data.
 11. The method of claim 10, wherein the adjusting step comprises determining a value of the signal correction parameter that optimizes the fitting metric.
 12. The method of claim 11, wherein the fitting metric is calculated using only nucleotide flows that result in nucleotide non-incorporation or single nucleotide incorporations.
 13. The method of claim 1, further comprising performing a base calling analysis of the signal data using the adjusted signal correction parameter.
 14. The method of claim 1, wherein the set of nucleotide flows is a first set of nucleotide flows and the sequence information is a first sequence information, and further comprising: applying the phase-state model using the adjusted signal correction parameter; calculating, using the phase-state model, predicted signals resulting from a second set of nucleotide flows that includes nucleotide flows that are not in the first set of nucleotide flows; making base calls by comparing the signal data to the predicted signals; and obtaining a second sequence information about the polynucleotide strand, wherein the second sequence information includes sequence information not contained in the first sequence information.
 15. The method of claim 14, further comprising repeating steps (d) through (g) using the second sequence information to obtain a further adjusted signal correction parameter.
 16. The method of claim 3, wherein the region is a first region and wherein the phase-state model is adjusted for a signal droop rate that is obtained by a method comprising: receiving signal data relating to chemical reactions in a plurality of wells within a second region of the array, wherein the plurality of wells includes the well containing the polynucleotide strand, wherein the second region is the same or different from the first region; calculating a set of averaged signal values from the signal data; and determining a region-wide signal droop rate by fitting a signal decay function to the set of averaged signal values.
 17. A non-transitory machine-readable storage medium comprising instructions which, when executed by a processor, cause the processor to: (a) receive signal data relating to chemical reactions resulting from a flow of nucleotides onto an array containing multiple wells, at least one of said wells containing a polynucleotide strand; (b) determine, using the signal data, sequence information of at least a portion of the polynucleotide strand; (c) construct a phase-state model for a set of flows that encompasses the sequence information, wherein the model includes a signal correction parameter and wherein the model is stored in a computer memory; (d) calculate, using the phase-state model, predicted signals resulting from the set of nucleotide flows; (e) compare the predicted signals to the signal data; (f) adjust the signal correction parameter of the phase-state model based on the comparison of the predicted signals to the signal data; and (g) store the adjusted signal correction parameter in the memory.
 18. An apparatus comprising: a machine-readable memory; and a processor configured to execute machine-readable instructions, said instructions which when executed cause the apparatus to: (a) receive signal data relating to chemical reactions resulting from a flow of nucleotides onto an array containing multiple wells, at least one of said wells containing a polynucleotide strand; (b) determine, using the signal data, sequence information of at least a portion of the polynucleotide strand; (c) construct a phase-state model for a set of flows that encompasses the sequence information, wherein the model includes a signal correction parameter and wherein the model is stored in the memory; (d) calculate, using the phase-state model, predicted signals resulting from the set of flows; (e) compare the predicted signals to the signal data; (f) adjust the signal correction parameter of the phase-state model based on the comparison of the predicted signals to the signal data; and (g) store the adjusted signal correction parameter in the memory.
 19. The apparatus of claim 18, further comprising: a plurality of reservoirs comprising different nucleotide reagents; and a flow chamber for receiving the nucleotide reagents.
 20. A method of sequencing a polynucleotide strand contained in a well of an array having multiple wells, comprising: receiving signal data relating to chemical reactions in a plurality of wells within a region of the array resulting from a flow of nucleotides to the array, wherein the plurality of wells includes the well containing the polynucleotide strand; calculating a set of averaged signal values from the signal data, wherein the set of averaged signal values are stored in a machine-readable memory; determining a region-wide signal droop rate by fitting a signal decay function to the set of averaged signal values; and storing the region-wide signal droop rate in the memory. 